Stpm Trial 2009 Matht Q&a Kelantan

papercollection

I.

Using a lcebraic laws ofsct.

prove that I(A u B)" C]u[(A u B) " C']- B =A-B. [6 marb]

1 .Ii

2.

Find

3.

On the same diagram, sketch the graphs of y

3x dx ZTx' + 1

[4 marb]

=x :

I and Y - 2x - J.

[3 morks)

A region R is bounded by the two graphs and the y-axis. Find the area of region R.(7 marh] A solid is fonned by rotating the region R completely about y-axis. Find the volume ofthc sol id fonned. [5 marb]

4.

The functionfis defined by fIx)

={~. x ~ 3 . O.

x =3

(i)

Find !~T- fIx) . !~T- f<x) nnd hence dctennine if the functionfis continuous atx - 3.

(i i)

Sketch the graph off

[3 marb] [2 marb]

S.

Prove that. for aU yaJues of k. the line kx + y - 8k is a tangent to the hyperbola xy = 16. Hence, find the coord inates of the point of contact. (S marks]

6.

Sketch the graphs of

y= ....!-. and Y =I2x Ix l

llonlhesamediagram. Hence, solvc the

inequality IZx - I I> ~ .

[7 marb]

Ixl

Y- 2

7.

8.

Findthevaluesofx.yandzif

4<-1)

x is symmetric. 2: - 1

[5 marb)

Given thatf(x) ... f + a:/ + 8x + b, where a and b are constant. f '(x) - 0 when x z: -2. When 1(..1') is divided by x + 2, the remainder is 2. Dctennine the \'alues of a and b. (8) Show that the equationf(x) - 0 has on ly one real root and find the set of values of x such thatf(x} > 0 (b)

Express x + 4 in partial fructions . fIx) I

9.

[

x'

4..1'; 4

Given the matrices A -

P

(

7

Find the values of p. q and r.

I

-I) (I

2 and 2 - 3 - I I 4

[5 marb]

papercollection 2009

954/1

Hence. solve the simultaneous equations x+ y - z ::-2 2y + Z a.5.r - 9 7x - 3y - : - 14

1y

10.

Given that y

) I.

Express _ -1_ - as partial fraction. (r + I)(r + 2)

, showthat (I + X2)d dx'

z>

, .. 2..

Find also the value of lim

)

dx

[4 marks)

(4 marks)

I

,.J-

Hence or otherwise. find the value of

+ 3x~ = O

----

.:-=, (r + l )(r + 2)

(6 marks)

I

L --(r + l )(r + 2)

[4 marks)

........ . ... 4 1

12.

By using sketching graph. show that the upproximate value of the root can be derived by the

following Newton - Raphson iterative fonnula X .. .

1

=- X..

-

f(x,, ) . f(x. )

[4 marks]

Show that the equation Ian x "" 2.r has a root in the interval ( ; . ~). Hence. find the root of me equation for x e (~ .

[3 marks]

%). correct to three decimal places.[7 mor.tsJ

papercollection 1. Ex press 3sinO-cosO inthefonn Rsin{O-a) ,with

R > O. andOo
18 J

equation 3sinO-cosB = I . for 0° <0<360°.

2. By using the substitution y = vx. show that the general solution for the differential equations

~ = x + 2}' is dx

AX l

3x

=(x _ y) l

I5I

,where A is a arbitrary constant

3. In a chemica1 process to obtain substance B from substance A. every molecule of A is directly change to molecule of B. The rate of increment of molecule B at any instant arc directly proportional to the number of molecule A at that instant. Initially the number of molecule of A is /,1 and the number of molecule B at t minutes after the process began is x. By assuming x as the continuous variable. Sbow that the differential equation representing tht' chemical process is

~ = *(M dl

x), with k as a eonstant

If x "" Owhent = O.showthat x=M(1- e- b

)

18 I

Sketch the graph of x versus t.

4. Ship A moves witb vclocity ( 30~+ 40j) kmh - ' and ship B moves with velocity (15~+12j ) kmh -' . Initially . the position vector of ship B relative to ship A is (5 ~ -1-12;') km.

Calculate the magnitude and direction of ship A relative to ship B and the nearest distance bctwccp ship A and ship B 15 1

5. (a) If ! and

!!... are the position vectors of two points A and B respectively .and P is a poi nt

which divides AB in the ratio m : n. Show that P has the position vector

no +mb r = - - --m+n

(b) Find a vector of the equation oflhe straight line passing through the point A with position vCClOr a = 2 i +.i and is parallel to the vector ~ = - 5 ~ - 2 j

PJ

I' I

papercollection 6. lbe figure shows a quadrilateral which inscribed two isosceles triangles ABC and ADE with base AB and AE respectively. Each trianglcs has the base angles of75 o . Be and DE are parallcl and equal in lengths.

~,~----------------------~;

Show that (a)

LeBE = LBED =90'

[4 )

r31

(b) ACD is an equilateral triangle

7. The random variable X is normally distributed with mean • jJ • and variance 400 . It is known that I'(X rel="nofollow"> 1159) ,; 0.123 and P(X > 769) ~ 0.881 . Determine the range of the value of I'. 15 )

8. In a certain city . records for burglaries show that the probability of an arrest in one week is 0.42 . l bc pro bability o f an arrest and conviction in that week is 0.35. (a) Find the probability that a person arrested for a particular week for burglary will be convicted . (b) What is the probability that there wi ll be 3 weeks of successful arrested and conviction in o nc particular month . .[5 }

9. The continuous random variable X has probability dcnsity function given by

lex) =

{

k(x+I),

O';x S 2

2k

2SxSb.

()

otherwise

Where k and b arc constants. (a) Given that P(X > 2.5)

=.!.. find the val ue of the constants 6

(b) Sketch the graph ofthc probabi lity density function [(x). (e) Find 1'(1 .5 S X S 2.5)

k and b.

16)

131 12 1

papercollection 10. A discrete random variable X has a probability function

P(X) ={

k(9-x)',

x = 5,7,8

o

.

otherwise

With k as 0 constant. (0) DetU' m i~ the value of k. (b) Find the cumulative distribut ion f ooction F(X).ond s ketch the graph of F(X) (e) Find the mean and variance fo r X (d) Find E(4X-2} and Var{ 4X- 2)

[2 J [ 4) [3

J

[7 J

II. The successful sale of T-shirts in a local shop is bi nomially distributed with the probability o f

selling one T-shirt (if thc customer enter the shop) is 0.35. (a) If 12 customers visit the shop in onc particular day, find thc probability that at least two T-shirts were sold. (2 ] (b) CaJculate the least number of cu.~ome rs nceded in order the probability of getting at least one sale is greater than 99 %. r3 J 12. The fo llowing data are the handphones prices . to the nearest RM 100, of40 hand phones taken at random from a handphone shop, arrange in ascending order.

100 1000 1400 2 100 3300

400 11 00 1500 2300 3400

500 1100 1600 2400 3900

600 1200 1700 2500 4 100

900 1300 1700 2600 4500

900 1400 1800 3000 4500

1000 1400

1900 3000

4900

1000 1400 2100 3300 5300

(a) Display the data in a .templot (b) Find the median and interquartile nUlge. (c) Calculate the mean and the standard deviation. (d) Draw a box piotto represent the data (e) State the sJlape of the fTequency distribution and givc a reason for your answer.

Elfd 0/ questions

[2 : [4I r5 '

(3 I ,~ j

Stpm2009

mls: I

Us ing algebraic laws of set, prove that

papercollection

[(A v B) nCJv[(A V B) nC 'J- B = A -B .

[(Av B) n C JV[(AvB) nC 'J - B =[(A v B)nCJV[(A V 8) n C'ln

Solution:

= (A vB) n (CvC) n B'

B'

[6 mar.lsl

BI BI

= (A v B) n~n 8'

= (A v B) n B'

BI

= (AnB') v (BnB')

BI

=(AnB') v ; = (A n B')

BI

= A-B

BI

:; 2.

Find

Solution:

r

3x

i 2:.rr:i

rl 2J7:i 3x

14 mar';..]

dx

'f 3U


MI

,3

=f-du ,2

AIM I AI

3.

On the same diagram, sketch the graphs of y = ~ and y '" 2x - I. (1 marks I x+1 A region R is bounded by the two gmphs and the y-axis. Find the area of region R.[7 marh ] A solid is fonned by rotating the region R completely about y-axis. Fi nd the volume of the [5 morh]

sol id formed .

Solution:

5 -

- ----"---+..,f-,-=--'--+

y= ~

010101

Slpm2009

papercollection

mls:2

~=2x- 1 H I

S = 2x2+X - 1 2xl +x - 6 = O (2x - 3)(x+2) = 0 MIAI

Area -

, j...!...dx - 2 X~+ . !. X3X~ o x+ 1 2 2 2

, 9 = 5In [x+ IU -3 +

4

=

5 In ~ - ~

MIMI

AI

2 4

Vo lume =

MIAI

'n5 ')' dy +"3"I (3 "ily2" )' (3)

,.. If

25

10 )

M IA I

9

1 dy+-1t y Y 4 X1'"--+

1

25- 101ny+ y )' +-If 9 =If ( - -

Y

= ,,[( - ¥ =

4.

,

AI

- 10In5 + 5)-( - ¥ - 10In2 +2

~ + IOln ~ 4

4

H1

MI

AI

5

{~. x

(i)

';C 3. O. x=3 Find lim I(x) . lim I(x) and hence detennine iflhe function/i s continuous al x = 3.

(ii)

Sketch the g.. ph off

The function/is defined by I(x) =

..... ,.

. ....r

[3 marls) [2 marls]

Solution: lim lex) = lim x - 3 = I

. -.,.

.-d·

x- 3

lim /(x) = lim -

H 3

=_ 1

.... ,. .... ,- x - 3 I{x) "'$. lim I(x) Since .lim _ .,' .-0)'

:. lis not continuous at x = 3.

MIAI

AI

Stpm2009

papercollection

mls:3

0---

DIDI 5.

Prove that, for all va lues of k, the line /l!x + y = 8k is a tangent to the hyperbola xy::: 16. '-Icnee, find the coordinates of the point of contact. [5 marks1

Solutio n:

y=8k-.I". .I"(8k - 6) = 16 k'x' - 8h+ 16 - 0 (h - 4Xh - 4) = 0 (h - 4)' - 0 4

MIAI

:. x=!

MI

.:::) There is only one point of contact :. The line Itx + y "'" 8A: is a tangent to the hyperbola .ry "" 16.

AI

When x = ! . x=4k k

:. The coordinates of the point of contact =-

6.

(~.4.t )

Sketch the graphs of y = ~ and y ""

Ixl

12x -

BI

l ion the same diagram. Hence. solve the

inequality 12x - II> ~.

[7 mOTu]

Ixl

Solution: y

Ilx - II

010101

...!.. =f 2.1" - 11 Ixl

:::) .!. = 2x - 1 .I"

=> 2..' - .1" - 1 = 0

Slpm2009

mls:4

=>«- I)(2x+

papercollection

I )~O

I

MIAI

x : J,- 2'

For

12x- I I>~, the solution

D

{x:x<

- ~,x > I}

MIAI

4X- I)

y- 2 x' Find the values of x, y and z if 4x - 4 2x X is symmetric. ( Y y+z 2% - 1

7.

Solution: Xl = 4% - 4 ,, - 4x - l, x"'y+z. =><'- 4x + 4 - 0, => (x - 2)2 a 0, =>x=2, y = 4(2) - 1 = 7,z = 2 - 7 =-5

MIAI

MtAIAI

Given thall(x) - x' + ax' + 8x + b. where a and b an: constant. I '(x) - 0 when x = -2. When f(x) is divided by x + 2, the remainder is 2. Determine the values of a and b. (a) Show that the equationj(x} "" 0 has only one real root and find the set of values ofx such thatl(x) > 0

8.

(b)

Express %+ 4 in partial fractions. I(x)

I(x) - x' + ax' + 8x + b ['(x) - lx' + 2ax + 8 1 (-2) - 12 - 40 + 8 = 0 40 ~ 20 a- 5 Whena - 5. I(x) - x' + 5x' + 8x + b 1(-2) - -8 + 20 - 16 + b - 2 -4+b - 2 b- 6 :.I(x) - x' + 5<' + 8x + 6

Solution:

ax' +0:<' + 8x+ b 1(- 3) = - 27 + 45 - 24 + 6 = 0 .: :c - -3 i$ a rool o/the equationj(x) = O.

a) [(x)

Lel[(x) = x'+ 5<' + 8x + 6 = (x + 3)(<' + 2x + 2) When[(x) - O. (x + 3)(<' + 2x + 2) - 0 .T _ _

[5 marb)

3.x _ - 2±~ 2(1)

-2±~

x - -3. x - - 2-

The equalion n,x) - 0 has only one real rool,

X"

- 3.

Whenl(x) > 0 (x + 3)(..' + 2x + 2) > 0 But ,r2 + 2.t + 2::: (x + 1)2 + 1 > O. since (x + 1)2 > 0 for all real valuesof x

Stpm2009 mls:5 Hence (x + 3)(x 2 + 2x + 2) > 0 Whenx+3 > O::::)x > -3 The set of values ofx such that}{x»O is/x: x >-3). b) x+4 _

t

x+4

(x + 3)(x' + 2.<+ 2)

f(x) l..c

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x+4 A nx+C (x+3)(x'+ 2x+2) · (x+3) + (x'+2x+ 2)

x + 4 . A (x' + 2.r + 2) + (x + 3)(& + C) Substitutingx - -3: I =A(9 - 6+2)

A=.!. 5

Comparing the coefficients of~: 0

=A + B

B- -.!.

5

Comparing the constant terms: 4 .,. 2.'1 + JC

~

4=

+3C

5

.!.8.

3C -

5

C- ~ 5

I

I

6

.: A=S,B- -S andC - S x+4 = _ 1_ (x+3Xx 2 + 2x+2)

9.

_

S(x+)

Given the matTices A

(x - 6) 5(x l + 2.%+2)

c(~ ~ 1

1 -2 ) and

- 3 - I

(~

:) satisfy AB c BA

I

Find the values of p, q and ,..

[5 marks)

Hence, solve the simultaneous equations

x+ y - z =-2 2y +z o:Sx-9 1x - 3y - z = 14

Since AB " SA AB - BA "" nl, where n is a scalar a nd I is the 3 x J idenfity molrix.

( ~ 4~I] (~ ; !]= 1

-3

-I

1 5 r

nf

[5 morbI

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mls:6

q- 2 pq+22

J 3p+ 16+2r = nl 7- r

7q-14

9- r

-r J (n OJ

q- 2

7 0 pq+22 3p+16+2r = 0 n 0 9- r 0 0 n 7q - 14

p+ 10 = 0, q - 2 =

oand 7 -

p=-IO,q=2and r=7

A=(~ ~ ~IJ~(_:O 7 -3 - I

r = O. Notethatn ~ 2

~IJ

4

7

- 3-1

Also, AB :: BA "'" 2J x+y-z "' -2

2y+z = 5x - 9. -IOx + 4y+2z = - 18 7,- 3y-z c 14 The 3 simultaneous equations can be written as a matrix equation as follows

_~ ~:J(;) =(~~8 J

HO

Premultipllying with matrix B

BAUJ= B(~~8l

V

(~J =B( ~~8)

2[~} [~ ;JU~8JSloce/(~) £[~) =

(~)=[~) 2.r = 4, 2y c - 2 and 2z = 6 x == 2.y =- 1 andz ; )

10.

Given that y =

2y

. show that O + x2}d dx'

+ 3x~ = O dx

(4 marks]

mls:7

Stpm2009 Answer: y = -

x

-

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-,

( l + x1)1

, y( I + X2)2 - x

y(1 +x, )- i (.!.X2x) + (1 +x, )i !!l. = 1 2 dx

y ' + (1 u , )i !!l. = 1 dx 2y !!l. + (I +x, )i d'y + !!l.(.!.Xl u, )-i (2x)= O dx dx' dx 2 2 (I+xl)i d y + 2[2Y + (I + x2 x1 = O dx' dx

)-i

(I +X, )i d

1y

(/xl

+ 3( _ _ X_)~= O ! fix

(I +X1 )1

(I+z' / 'y + 3z!!l. = O dx' dx

II .

Express - -1- - as partial fraction. (r + IXr +2)

Hence or otherwise, find the value of

[4 mar,u]

._2. 1 L ---, ••• , (r+IXr +2)

. ,· 1.. I Find also the value o f lim ....... , ..... , (r + IXr +2)

L ----

Answer: Let _ _ 1_ _ = _ A_+ _ B_

Substitute

(r+ IXr + 2) (r+l) (r + 2) I - A(r+2) + B(r + I) r ... -2 I = - B, r '"" -I if .. I

(r + IXr + 2)

(r + l)

B :: - I

(r + 2)

'f __1_ _ ;; ,'f.... , (~~) , .... , (r + IXr + 2) r + 1 r +2 (_ I __ ~)+ (~ __I _ ) + (_ I __ ~) + n+ 2 n+ 3 n +3 n + 4 n + 2 n +3 (_ I __ ~) + n+ 4 n+ 5 (_ 1_ _ _ 1_ ) . 2n + 1 2n + 2

n +2

1 2n + 2

....... + (-,- _ _1_ ) + 2n 2n + 1

[6mar,u]

[4 mar,u]

S!pm2009

papercollection

mls:S 2("+1)("+2)

1

~_l..

IimL -+-I)(r-+2)

.40, ••• , (r

n

lim

·-2("+ 1)("+2) lim(_I-- _ I-)

..... n+2

o 12.

2n+2

By using sketching graph, show that the approximate value of the root can be derived by the

following Newton - Raphson iterative fonnula x •• , =x. - f(x. ) . f(x.)

[4 marks]

Show that the equation tan x = 2x has a root in the interval (~.

%).

[3 marks1

Hence. find the root of the equation for xe (~, ~), correct to three decimal places.[7 marks] Answer.

frx)

x~

Q(x". ,)

f{x.)]

x,.

01 Refer to the graph above, Q is the point of intersection of tangent at P and the x-axis. Gradien! ofll>e langen!.! P = f(x.) :. Equation of langen! . ! P is:y - j(x.) = f(x.) [ x - x.l A!Q.y= O.O - j(x.) =f(x.)[x", - x.l MIAI

:. x•• , = x. - f(x.) , f'(x.) f(x.)

~0

AI

li~tanx =1X!

MI

" :. The equation tan x = 2r has a root in the interval

j(x) - lanx - 2x f(x) =sec' x - 2 Letxo "" I,

XI = X(J -

!(xo) f'(x. )

(~ .

%).

AI

SIpm2009

papercollection

mls:9 x = 1_ tanl - 2 = 1.3 105 I

secl l - 2

X,

= 1.3105 _ tan(I.31 05) - 2(1.31 05) - 1.2239 sec'(1.3 105) - 2

X,

= 1.2239 _ tan(1.2239) - 2(1.2239) _ 1.1760 sec'(1.2239) - 2

x, = 1.1760 tan(1.I760) - 2(1.I 76O) = 1.1659

MIAI

MI

sec'(1.I76O) - 2 x, = 1.1659 tan(1.I659) - 2(1.1659) = 1.1656 sec' (1.1659) - 2 :. x = l.l66

MI AI

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G

rz. .~

,c:; .. ({! - c<) (,.",'0 0 - ((.

R

3 I

(-!!

0( 0(

fin

3

~

l {j. 43

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Co< ( ,'1 \

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papercollection Vx ·

: V

=

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dv

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':t

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d,( ( rlV

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\" [ X( r - v) 3J

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e. li

X( I - V)J =

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)( ( 'X.-;'_'L/)

J

~ (" " (1 ) ,

~

A

- fj,

xJf>

(,( 'J) ' - A ('

II ~

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(/}

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I~.,f

~!

i

k(t11 -)( 0

::

II

I' f

/11", ' . /H. · i ,: ,1<

c1{

.Ti

=( k

JM -><

..J t .. /.'

- \

- l fI((v1 -~ )

/ ,., (IIA) =C

~ - Kf

M -

M

>(

')( =

M -

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M

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k ( M - .,.. )

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\ k-(

M e- I,,!

( I -

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J /<;,,-;);;" "0\ / '

n]r« -f;<.

r> ,-

rJ ~

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o--' A:

q

Oii ;- ~

p n --'

OJ>

~d

=

?

OP 0 "

( p lYh 1,,,,, Vf CW- )

(3-:" 01 n

-=

o (r -4 ) = Ilr _ n tl

t

n r

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(Yl r-

N)b

-

fY'

h

=-

~

-or

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o~

b _ r

rnr

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f"It\~ fY'b

.-.( IVl-ln ) f'"

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--> PB (b - rl

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p(i I 0 5

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r'YI

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